Peter G.Wolynes
1570
Osmotic Effects Near the Critical Point Peter G. Wolynes* Department of Chemistry, Harvard University, Cambridge, Msssachusetts 02 138 (Received March 3, 1976) Publication costs assisted by Harvard University
Near the critical mixing point of binary solutions, anomalous osmotic effects can occur. These effects involve the interaction of the microscopic structure of the mixture near a wall and the nonlinear coupling of the composition and momentum of the fluid. The Onsager principle relating the osmotic and inverse osmotic effects is verified. Diffusion through a capillary, which has a contribution from the same causes as the osmotic effects, is shown to provide a particularly simple example of mode-mode coupling theory.
I. Introduction Hydrodynamic quantities which by symmetry are uncoupled in a bulk fluid may be coupled together in the presence of a boundary. In a fluid in which there are no pressure gradients, concentration gradients cannot linearly couple with the m0menturn.l Near a wall, however, concentration gradients can force fluid motion. This phenomenon in dilute gas mixtures is referred to as diffusion slip (or creep).2 It is an effect which becomes noticeable when the characteristic dimension of the fluid flow becomes comparable to a mean free path. This phenomenon in solutions is very familiar as osm o s i ~If. ~two solutions of different composition are connected by a thin capillary or porous plate, fluid will flow between them until the osmotic pressures are equalized. An inverse coupling exists. If a dilute gas mixture is bled through a small opening demixing results. The composition of a fluid forced through a porous plate changes. This is inverse osmosis. In measurements of viscosity near the critical point of binary mixtures by the capillary method, hysteresis has been observed.* Also Leister and co-workers have noted separation of the mixture when forced through a capillary near the critical point.5 In this paper we discuss a simple theory of osmotic effects near the critical consolute point of binary mixtures. This theory is analogous to the theory of electroosmosis of dilute electrolyte solution^.^ Its essential features involve the microscopic structure of the solution near the capillary walls and the nonlinear couplings of the momentum and concentration which are thought to give rise to the anomalous behavior of the viscosity and diffusion constant near the critical point. This theory does not seem to be sufficient to explain the observation of Leister et al., but is of at least pedagogic value. We examine explicitly the Onsager relation of the osmotic and inverse osmotic effect and will see how a relation between the nonlinear coupling coefficients and the equilibrium properties of the system is implied by the Onsager symmetry. This relation is exactly that predicted from microscopic theories of the nonlinear coupling coefficients. Also diffusion through a capillary (or porous plate) is considered and will be seen to be a particularly simple manifestation of mode-mode coupling theory.
11. Theory T o simplify the calculations we consider the following geometry (Figure l).Two very large vessels are connected by a rectangular capillary of large length 1. The capillary has a width w which is much larger than its depth d. In the following we will consider the case where d is larger than the correlation length $. = K - ~ . We assume that one component is preferentially adsorbed on the glass walls of the capillary. Let us define the concentration variable c(X) as the difference in concentration of one of the components from its average value in the bulk medium. The concentration at the wall is determined by the difference in the standard chemical potential of the substance adsorbed on the wall and the standard chemical potential in the bulk medium, Since the standard chemical potential in the bulk medium is nonsingular at the critical point we take c(X) at the wall to be a constant co # 0. Theories of inhomogeneous systems near the critical point, such as the van der Waals theory of surface tension: indicate that c(X) satisfies the Ornstein-Zernike equation:
* Present address: Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Mass. 02139.
Assuming stick boundary conditions on the capillary walls u (0) = u(d) = 0 the velocity profile7 is
The Journal of Physical Chemistry, Vol. 80,No. 14, 1976
0%=
(1)
K2C
Since we are neglecting end effects the concentration profile satisfies the equation d2cJdt = K
~ C
(2)
with the boundary conditions c(0) = c(d) = co the concentration profile is therefore c ( z ) = co
cosh K ( Z - dJ2) cosh ~ d / 2
(3)
If fluid is forced through the capillary this concentration profile will be convected along. The material emerging from the capillary will be of a different composition than the bulk medium. This is an inverse osmotic effect. A pressure gradient along the capillary will induce fluid motion. At steady state the velocity will satisfy the steady Navier-Stokes equation for an incompressible fluid.
v.v=o
(4b)
1571
Osmotic Effects Near the Critical Point
illary must cause the fluid to flow. This osmotic effect is due to a nonlinear concentration dependent stress tensor for the fluid. While symmetry requires that the velocity cannot linearly couple to the concentration it is possible to have nonlinear stresses proportional to the products of the concentration gradients: u = a(Vc)(Vc) (11)
c +ac
C
j'dS
P+AP
I
I&
/-
This type of stress tensor was first introduced by Korteweg in his theory of hydrodynamic capillary phenomena! It is also used in mode-mode coupling theories of the anomalies in transport coefficients of critical mixtures.gJ0 Because of the concentration profile in the capillary, a concentration gradient along the capillary can force the fluid motion by this nonlinear stress. The Navier-Stokes equations with this nonlinear stress are:
I / /
P I
C=C,
cosh K (Z -d/2) cosh ~ d / 2
Flgure 1. Diagram of hypothetical capillary for measuring osmotic ef-
fects.
(5)
J=
s
cv dfl = w
x co
xd
dz
-
cosh K ( Z d/2) -A- p[ ( 1 z-;) c o s h ~ d 2 1 29
d
2
-,1 d2 (7)
If d is much greater than the correlation length the concentration profile decays exponentially away from the wall at z = 0 and we can replace c and v by the approximate form c = cOe-KZ
v(z) =
(8a)
(124
v.v=o
(12b)
Considering the case without a pressure gradient the flow is described by the equation
The total flow through the capillary is the integral of this velocity over the cross section of the capillary I = s v d f l = - -wd3 A p 127 1 This is Poiseuille's formula for flow through a rectangular capillary. The concentration current j is the product of the velocity and the concentration C. The concentration flux through the capillary is
vV2v = -vp - a(V2c)Vc
d2v
Ac
where Ac/l is the concentration gradient along the capillary. Imposing stick boundary conditions on the walls of the capillary the velocity profile is
Using the concentration profile (3), we find that the total fluid flux due to the concentration gradient is
When the capillary diameter is greater than the correlation length, this is a
I = - cowd 9
(T)
It is useful to rewrite this expression in terms of the thermodynamic force, the chemical potential gradient:
zd A p
a ac AW I=-cOwd 9 (--)-=A 1 uc-
29 1
for z near 0 and analogous formulae when z approximately
d. Thus J is
J = 2 w i m d z c o e - K z ( - rzd - )A p
1
The Onsager reciprocal principle then requires the equality coefficients in eq 9 and eq 17:3 A,, = A,,
9 1 a!
This concentration current diverges quite strongly near the critical point. However, we should recall that the derivation only holds when d is greater than a correlation length. When d is less than a correlation length the concentration current should level off to a constant value. The change in concentration of the emerging fluid is
AP
(18)
This is true only if the nonlinear stress tensor coefficient satisfies the relation: (19)
This relation is in fact satisfied. Kawasaki has shown by microscopic calculations that a is related to the h dependent susceptibility X k : l 0 a = kgT-
Thus demixing can become quite large when the diameter of the capillary is comparable to the correlation length. Since this inverse osmotic effect exists, nonequilibrium thermodynamics requires that a concentration gradient along the cap-
We also know that hBTXk=o-l is the thermodynamic derivative awlaC. Thus (21) The Journal of Physical Chemistry, Vol. 80, No. 14, 1976
Peter G.Wolynes
1572
If the spatial correlations are of Ornstein-Zeinike form, as we have assumed in the derivation of the transport coefficients, Xk is proportional to ( k 2 k 2 ) - l . Thus we have
+
which is the same as eq 19. Thus we see that the Onsager relation is satisfied in a fairly nontrivial manner. If there is a concentration gradient along the capillary a concentration flux will be established. One contribution to the flux will be due to diffusion. The diffusive flux will be Ac 1
J D = Dwd -
(23)
The diffusion constant vanishes at the critical point. Mode-mode coupling theory indicates that it vanishes much more slowly than the conventional theory of critical slowing down would indicate and gives the relationlo
markedly noncritical composition throughout the capillary and because the mode-mode coupling processes in such a small capillary will be different from those in a bulk fluid because the fluctuations must satisfy boundary conditions on the capillary walls. For these reasons we cannot trust our derivation for the total concentration flux through a capillary when it is smaller than a correlation length. When d is greater than E the convective contribution is only a "surface" term but it can be comparable to the diffusive flux:
To estimate this ratio we use eq 22 for a! and use the Ornstein-Zernike relation between the correlation length 6 and the short-range correlation length 10 which is of the order of an intermolecular d i ~ t a n c e : ~
E = ldc,hBT/(ap/ac)
(29)
where cc is the critical concentration in particles per unit volume. If co is of the order of c, which is of the order of one particle per lo3 we obtain In addition to the diffusive flux there is a convective contribution. As we have seen, a concentration g r d i e n t interacting with the concentration profile of the capillary can force fluid motion. The moving fluid can then convect the concentration profile, leading to a net transport of concentration. This process is directly analogous to the mode-mode coupling process which leads to the breakdown of the conventional theory of the vanishing near the critical point of diffusion constant. In the bulk fluid, an imposed concentration gradient interacts with a concentration fluctuations to give rise to fluid motion through the nonlinear stress tensor. The resulting fluid flow transports the concentration fluctuation and causes the net concentration flux to be larger than in the absence of these nonlinear processes. In the transport through a capillary the concentration profile plays the same role as the concentration fluctuation does in the case of diffusion in the bulk fluid. T o find the convective flux we use eq 3 for the concentration profile and eq 14 for the velocity profile due to a concentration gradient: Ac {cosh2K Z - cosh K Z ) a dz - - co2 J, = cu dl = w cosh2 ~ d / 2 1 1 1
s
= -w
a B
-co2
1
cosh2Kd/2
1% 2',
-t- sinh Kd
When d is much greater than the correlation length this becomes AC 1
J , = f w Ly c O ~ K - ~ 17
(26)
This "surface" contribution to the flux through the capillary diverges near the critical point. When d is much less than the correlation length the convective contribution to the flux vanishes:
The total flux through the capillary is the sum of the diffusive flux and the convective flux. I t is clear that when d is much less than ( the diffusive contribution will not be correctly given by eq 23 with the diffusion constant from eq 24. The diffusion constant will be in error both because the The Journal of Physical Chemistry, Vol. 80, No. 14, 1976
JJJD
N
E2/10d
(30)
Thus the ratio of convective to diffusive flow depends on the ratio of the correlation length to the geometric mean of the capillary diameter and the short-range correlation length. This ratio can be large even when d is larger than E. The osmotic effects discussed in this paper do not explain the demixing phenomena observed by Leister et al. In their experiments the capillary diameter is about .2 mm and in their measurements closest to the critical point the correlation length is about mm. Equation 10 indicates that the change in concentration of the fluid is only about a ten thousandth of a percent. This is too small to explain the observed demixing. Leister notes that the demixing is a nonlinear effect, increasing with the rate of fluid flow.5 This is of course not indicated by the linear theory considered here, but extensions of the basic effects, which we have studied, to include nonlinear phenomena and treating the critical fluctuations may perhaps lead to a better explanation of the experimental observations. We should note also that the linear effects considered here are quite observable. Order of magnitude calculations indicate that osmosis through porous media, such as millipore filters, can be quite considerable and that the convective contribution to concentration flow can be comparable to that due to diffusion. Acknowledgment. I thank Professors J. M. Deutch and Roy G. Gordon for support. References and Notes (1) R. H. Trimbleand J. M. Deutch, J. Stat. Phys., 3, 149 (1971). (2) H. A. Kramers and J. KistemaFer, Physica, I O , 699 (1943). (3) S. R. de Groot and P. Mazur. Non-equilibrium Thermodynamics", North Holland Publlshlng Co., Amsterdam, 1962.
(4) J. V. Sengers in "Proceedings of the International Summer School of Physics Enrico Fermi", Varenna, Italy. 1970, M. S. Green, Ed., Academic Press, New York. N.Y., 1971: J. Brunet and K. E. Gubbins, Trans. Faraday SOC.,65, 1255 (1969). (5) H. M. Leister, J. C. Allegra, andG. F. Allen, J. Chern. Phys., 51,3701 (1969). (6) B. Widom in "Phase Transitions and Critical Phenomena", Vol. 2, C. Domb and M. S. Green, Ed., Academic Press, New York, N.Y., 1972. (7) L D. Landau and I. M. Ltfshitz, "Fluid Mechanics", Addison Wesley, Reading, Mass., 1959. (8) C. Truesdell and W.Noll, "Handbuch der Physik". Vol. IlV3, S. Flugge, Ed., Springer-Verlag, Berlin, 1965, p 513; D. J. Korteweg, Arch. Neerl. Sci. Exactes Nat., Ser. 2, 6, 1 (1901). (9) M. Fixman, J. Chem. Phys., 47, 2608 (1967) (IO) K. Kawasaki, Ann. Phys. (N.Y.),61, l(1970).
